I was surprised to find out that the continued fraction expansion of $e$ is $[2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...]$. Even though $e$ is transcendental, this expansion is very nice.
We can describe it fairly succinctly by saying $a_0=2, a_1=1$ and for $n \geq 2$:
If $n \equiv 0$ or $n \equiv 1 \mod 3$ then $a_n=1.$
If $n \equiv 2 \mod 3$, $n+1=3k$, then $a_n = 2k$.
Now my question is: how can we characterize those numbers which have a continued fraction expansion that cannot be described by a finite formula?
By this I mean a finite number of statements which, taken together, describe every $a_n$.